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An investigation into mechanical properties of the nanocomposite with aligned CNT by means of electrical conductivity
Composites Science and Technology 188 (2020) 107993
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Composites Science and Technology journal homepage: http://www.elsevier.com/locate/compscitech
An investigation into mechanical properties of the nanocomposite with aligned CNT by means of electrical conductivity Dayou Ma , Marco Giglio , Andrea Manes * Politecnico di Milano, Department of Mechanical Engineering, via la Masa, 1, 20156, Milan, Italy
A R T I C L E I N F O
A B S T R A C T
Keywords: Nano composites Electrical properties Interface Mechanical properties Finite element analysis (FEA)
In the present study, a novel modelling approach based on electrical properties was proposed to replicate the mechanical behaviour of aligned carbon nanotube/polymer nanocomposites. Firstly, an electrical analytical model with Monte-Carlo method involved was established and validated by accurately predicted electric con ductivity. The microstructure of the nanocomposite was then determined according to the electric property. Subsequently, a large-scale representative volume element model based on the predicted distribution of the carbon nanotubes was built to replicate the mechanical response of the nanocomposite under tension, which can be validated by existing experiments. To consider the crystalline structure of the matrix, two cases on the nanocomposites with crystalline and amorphous polymer were investigated, locating their difference on the bonding condition of the interface between CNT and matrix. Results evidenced that the electrical properties of nanocomposites can be used to identify the internal microstructure of nanocomposite. Moreover, the effects of the loading direction, the interfacial strength and the weight fraction were studied by numerical models. The reinforcement effect of the carbon nanotubes was significant when loaded along the aligned direction, but the effect was limited in the other directions. The modulus and the strength of nanocomposite were improved by the increase of the weight fraction of CNTs, while the increase of interfacial strength improves the strength of nanocomposite along CNT-aligned direction significantly, but had negligible effect on its modulus.
1. Introduction Nanocomposites have attracted great attentions in recent years due to their unique properties and potential applications. Among them, a polymer reinforced by carbon nanotube (CNT) is regarded as one of the most promising novel composites, since the reinforcement with CNTs not only improves the mechanical properties but also makes the com posites conductive, which provides a novel approach to investigate the mechanical behaviours of nanocomposites based on electrical properties. Considering the numerical and analytical analysis on the electrical properties of nanocomposites, the theory of electrical treeing and breakdown was proposed to describe the conductive mechanism of nanocomposites: a current path can be created among the CNTs without direct connections, which attributes to the tunnelling effect among CNTs [1]. Based on this, Tanaka [2] introduced a multicore model with the CNTs surrounded by multi-interface to describe the interaction zone between the CNTs and the matrix and it provided the proper electrical property of the nanocomposite. Li et al. [3] proposed a formula to
express the resistance of the tunnelling effect, which accelerated the development of the electrical analytical model. Meanwhile, Landauer-Buttiker (L-B) model, proposed to predict the resistance of the nanomaterials, and Monte-Carlo method, used to model the stochastic events, were proved to be helpful in electrical modelling [4,5]. Based on the aforementioned methods, the reliable electrical analytical model can be built for the investigation on the effects of the CNTs’ dimension, waviness and distribution on the electrical property of nanocomposites [6,7]. Briefly, all of these investigations indicate that a numerical methodology is able to accurately predict the electrical properties of nanocomposites. Regarding the mechanical properties of CNT nanocomposites, ex periments were mainly focused on tensile tests [8–16]. According to these experimental data, varying strength and modulus were reported in different studies but all confirmed the improvement of the modulus by the addition of CNTs. Besides, both strength and modulus can be rein forced when aligned CNTs are spread inside the polymer. As for the simulations, there are mainly two methods: a molecular dynamics (MD) simulation and a finite element (FE) method. The MD simulation is able to consider the chemical bonding and the related microstructure of the
* Corresponding author. E-mail addresses: [email protected] (D. Ma), [email protected] (M. Giglio), [email protected] (A. Manes). https://doi.org/10.1016/j.compscitech.2020.107993 Received 26 June 2019; Received in revised form 31 December 2019; Accepted 3 January 2020 Available online 8 January 2020 0266-3538/© 2020 Elsevier Ltd. All rights reserved.
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Nomenclature
Electron mass kg Polar angle � Maximum polar angle (position angle) � Azimuthal angle � The uniform random number in the interval [0,1] Intrinsic resistance of the CNT kΩ Contact resistance between the two closest CNTs kΩ Strength along the longitudinal direction MPa Strength along the transverse direction MPa Volume fraction of the nanocomposite Volume fraction of the percolation threshold Volume of object in percolation theory m3 Excluded volume m3 Transmission probability to breakdown the matrix between two CNTs ΔE The height of the barrier between polymer and CNT eV h Quantized resistance (� 12.9054) kΩ 2 2e ðxi ; yi ; zi Þ Coordinates of the start point of the CNT in model μm
me θ θmax φ RAND RCNT Rcontact SLD STD V Vc Vo Vex T
Symbol Description Units μ Poisson’s ratio D The diameter of the CNTs nm d Distance between two CNTs in the model nm dcutoff The cut-off distance in the tunnelling effect nm dtunnel The tunnelling characteristic length nm dvdW Van deer Waals separation distance Å e Electron charge E Elastic module GPa g Electric conductivity of the nanocomposite S/m gCNT Electric conductivity of the CNT S/m gpolymer Electric conductivity of the polymer S/m h The Planck’s constant (2πℏ ) J⋅s l The average length of the CNTs nm lij The length of two contact nodes i and j along one CNT μm Lx;y;z The length of the electrical RVE along the x-, y- & zdirection μm M Number of tunnelling channels
CNT-matrix debonding damage mechanism, which depends on the shear strength of the interface. Additionally, the failure mechanism of in terfaces also depends on the crystalline structure of the matrix [32]. Among these experimental studies, however, precise equipment is required and data are affected by a not negligible spread. Therefore, improvements about modelling methods are expected for a compre hensive understanding of mechanical response of nanocomposites. In present the study, in order to replicate the mechanical properties of the nanocomposite, a novel approach is proposed to model its microstructure based on the electrical conductivity. Because the microstructure can determine both the electrical and the mechanical behaviour of the polymer/Carbon nanotube nanocomposite, such an approach could be useful to predict one behaviour when knowing the other, especially when the nanocomposite is regular, such as aligned CNT employed in present study. Firstly, an electrical model, based on the previously introduced model in combination with the Monte-Carlo simulation, was built to replicate the electric conductivity of the nano composite by randomly distributed CNTs in the matrix. After the vali dation of this model, the microstructure of the nanocomposite with aligned CNTs can be obtained. Subsequently, a large-scale FE model was created to replicate the mechanical properties of this nanocomposite based on the results from the electrical model. The process is described in Fig. 1. Herein, the method was validated through the experimental data about tensile tests on the aligned-CNT/PET nanocomposite form [14] (named Case-1) to present the reliability of the proposed approach. Considering the crystalline structure of the polymer, one additional case, aligned-CNT/PSF nanocomposite from Ref. [16] (named Case-2), is also involved in the validation since PET is always regarded as a semi-crystalline polymer while PSF is a typical amorphous one. More over, the effect of the interfacial strength, the loading direction and the weight fraction is also discussed in a numerical framework.
nanocomposites. Therefore, the investigation of the interface between the CNTs and the matrix is always carried out in an MD simulation [17, 18]. Additionally, the effect of the CNTs dimension [19], posture [20] and distribution [21] on the mechanical property of nanocomposites has been investigated in this way, and their significant effects were proved. However, a long calculation time [20] and considerable parameters related to chemical bonds [22] cannot be avoided, hindering its wider application. Considering the FE method, the generation of a proper mesh is complex due to the dimensional gap between the CNTs and matrix. However, the FE method is still widely used on nanocomposites in the community due to its reliable modelling of complicated structures under various loading conditions and the user-friendly simulation environ ments. For instance, Chen and Liu [23] created a representative volume element (RVE) on the microscale to investigate the stiffness of nano composites. Arora et al. [24] studied the mechanical behaviour of nanocomposites along different directions and suggested to consider the material reinforced with aligned CNTs as a transversely isotropic ma terial. Also, Shin et al. [25] combined both the MD simulation and the FE method, which provided a new insight for the numerical study of nanocomposites. All of these existing studies, however, still have some limitations with regards to the modelling of the nanocomposites’ mechanical be haviours: the accurate microstructure of the nanocomposites and the failure mechanism of the interfaces between the CNT and matrix. Only one/a few CNT(s) can be contained in the numerical model because of computational costs [17,18,20–22], or random generation was employed to build the CNT system inside nanocomposites [4,6,7,26]. Although the Monte-Carlo simulation can provide good results through the random generations, extensive calculations are inevitable. Efforts have been made on modelling the microstructure of nanocomposites according to the images of sample cross-sections [27,28]. However, the accuracy might be affected due to the possibility that the real structure cannot be represented by a single cross-section and the cost of this technology presents a further drawback. Therefore, large-scale model ling, especially containing a precise microstructure, is still a challenge in the simulation of nanocomposites [29]. As for investigation about the failure mechanism of interfaces between the CNT and the matrix, Yu et al. [30] combined a loading device and scanning electron microscopy (SEM), while Cooper et al. [31] studied the tension response of the nanocomposite with post analysis by transmission electron microscopy (TEM). They reported that the “pull-out” failure of CNTs is the dominant
2. Electrical simulations 2.1. Modelling strategy In the present study, all CNTs are assumed to be straight to model aligned CNTs, therefore a line segment can be used to model the CNT. Considering the random distribution of the CNTs inside the matrix, the start point of the CNT ðxi ; yi ; zi Þ, as shown in Fig. 2, is generated ac cording to Eq. (1). 2
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Fig. 1. flowchart for present method.
Fig. 2. A schematic diagram of the CNT inside the nanocomposite.
perfect circle with the diameter D. lij is the length of two contact nodes i and j along one CNT. The contact resistance, Rcontact , caused by con tacting between CNTs and the tunnelling effect. According to L-B model [34], Rcontact is defined by Eq. (5), where 2eh2 is equal to the quantized resistance (� 12:9054kΩ). Moreover, the transmission probability to break down the matrix between two CNTs, T, is determined by Eq. (6) based on Wentzel-Kramers-Brillouin (WKB) approximation [35]. With the Pauli exclusion principle [36], the closest distance between two CNTs, d, should be no less than the van deer Waals separation distance, dvdW . So, if d is smaller than dvdW , it should be replaced by dvdW , which is used to model the resistance due to contact between the CNTs. When d is greater than dvdW , the real distance is equal to (d D) considering the diameter of the CNT, i.e. the tunnelling effect is built in this way. But the tunnelling effect disappears if the distance is beyond the cut-off distance, dcutoff , then there is no effective current path any more. Moreover, dtunnel pffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the tunnelling characteristic length defined by dtunnel ¼ ℏ= 8me ΔE.
(1)
Θi ¼ LΘ � RANDðΘ ¼ x; y; z:i ¼ 1; 2; …Þ
A CNT built as a line segment and the coordinate system generated based on the CNT are shown in Fig. 2, θ is the polar angle, between the CNT and the z-axial, defined by the maximum polar angle θmax (Eq. (2)), while the azimuthal angle φ, the angle between x-axial and the projec tion of the CNT on the x-y plane, is generated inside [0, 2π] randomly (Eq. (3)). The θmax represents the position angle in present study, which determines the maximum polar angle of all CNTs. If θmax ¼ 0, the nanocomposite is assembled with all CNTs aligned along one direction. If θmax ¼ 90� , the nanocomposite can be regarded as an isotropic ma terial, which indicates the totally random distribution of aligned CNTs in the matrix. cosθ ¼ ð1
(2)
cosθmax Þ � RAND þ cosθmax
(3)
φ ¼ 2π � RAND
As for the length of each CNT, an average length l of all CNTs was used in the present study. The average length can always be obtained by atomic force microscopy and the length of the CNTs should follow the Weibull distribution [33]. Subsequently, the coordinates of the end point of ith CNT can be described as (xi þ lsinθi cosφi ; yi þ lsinθi sinφi ; zi þ 0
RCNT ¼
4lij
π gCNT D2
Rcontact ¼
lcosθi ). Moreover, if the end point is out of the boundary of electrical RVE, the CNT would be cut directly by the boundary. In this way, a random CNT network can be generated in the electrical RVE. Two types of electric resistance existing in this CNT system contribute to the total resistance of the nanocomposite. One is the intrinsic resistance of the CNT, RCNT , and the other is the contact resis tance between the two closest CNTs, Rcontact . RCNT can be obtained through Eq. (4), where the cross section of the CNT is regarded as a
h 1 2e2 MT
� � dvdW exp ; 0 � d � D þ dvdW dtunnel T¼ � � > d D > > : exp ; D þ dvdW � d � D þ dcutoff dtunnel 8 > > > <
(4) (5)
(6)
In the present study, the distance between every two CNTs was calculated through the methodology mentioned in Ref. [37] providing a matrice of distance between every two CNTs in the network. Then, the 3
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matrices of all potential resistances can be calculated by Eqs. (4)–(6). Subsequently, a searching algorithm was employed to obtain the effec tive current method in the CNT system and all of the electric resistance from the effective current paths are regarded as being connected in parallel. Additionally, the Monte-Carlo simulation was involved in this electrical model for each case. The above-mentioned process was per formed in MATLAB®.
provided a possible method to link the percolation threshold to the aspect ratio of the CNT based on the statistical percolation theory. Meanwhile, the DPD (dissipative particle dynamics) method coupled with Monte-Carlo simulations is also able to replicate the percolation threshold [43] but requires time-consuming calculation. Herein, we employed the statistical percolation theory to obtain the percolation threshold and compared it with the existing data to validate the present electrical model. Based on the statistical percolation theory with t ¼ 1:414, the percolation threshold is approximately 0.62 vol%. With these parameters, the linear relationship of the logðgÞ and logðV Vc Þ is pre sented in Fig. 3, as a validation. The value of the percolation threshold predicted in the present work stands in good agreement with the threshold predicted in Refs. [7,28], while the value theoretically pre dicted by the aspect ratio of CNTs [42] is 0.631 vol%. Therefore, the present electrical model was validated and can be used in the next stage. In the present study, the electrical model introduced in this section was employed to obtain the position angle θmax exploiting a reverse engi neering approach.
2.2. Validation of the electrical model To validate the electrical model, the electric conductivity of a multiwall CNT (MWCNT) distributed in the polymer was calculated. All the data used in present model were taken from Refs. [7,38]: gCNT ¼ 5� 103 S/m, gpolymer ¼ 10 12 S/m, ΔE ¼ 1 eV, l ¼ 5 μm, D ¼ 50 nm, M ¼ 460, Lx ¼ Ly ¼ Lz ¼ 5:5 μm, θmax ¼ 2π , dvdW ¼ 3:4 Å and dcutoff ¼ 1:4 nm. The increasing trend of conductivity with volume fraction is shown in Fig. 3. The results obtained from the present model correspond to the existing experimental data [7,28]. The electrical conductivity steadily increases while its growing rate decreases with the increasing weight fraction of the CNTs, indicating the gradual disappearance of the tunnelling effect. As the tunnelling effect plays the dominant role in generating the effective current when the volume fraction is low, the electrical conductivity is sensitive to the number of the CNTs in this region. On the contrary, the increase of the electrical conductivity is no longer significant after reaching the percolation threshold, which is the saturation point of the tunnelling effect, as shown in Fig. 3. There are different theoretical methods to predict the percolation threshold of nanocomposite. The theory related to the concept of excluded volume (Vex ), which is determined as the volume around the centre of one object which doesn’t allow others to penetrate, is reported effective [39]. The percolation threshold can be expressed by the excluded volume as Vc � Vo =hVex i [40], where h i is the average calculation in a random-distributed system, Vo is the volume of the object and Vc is the percolation threshold represented in the volume fraction. In addition, based on the data of the electric conductivity, the statistical percolation theory can also provide accurate results, which can be expressed by describing the relationship between the existing data about the volume fraction (V) and the electric conductivity (g) using g ¼ A⋅ ðV Vc Þt . The parameters A and t are defined by the data when the logðgÞ and logðV Vc Þ are linear-related. However, t is proven to be related with the material experimentally [41], while A can also be calculated through the conductivity of the nanofillers [42]. Moreover, the work of [42] also
2.3. Simulation results In this section, the position angle θmax of the polyethylene tere phthalate/MWCNT (PET/MWCNT, Case-1) and polysulfone/MWCNT (PSF/MWCNT, Case-2) were determined based on the electric conduc tivity reported in Refs. [14,16], respectively. For Case-1, the CNTs were dispersed into PET and then aligned along one direction (marked as z-direction in the present work) during the manufacturing process. Three weight fractions of CNTs, 1.0%, 2.0%, and 4.0%, were investi gated. Relative properties of the MWCNT (Nanocyl® 7000) [11,44] and PET (Polyclear F019) [10,14] are listed in Table 1 and Table 2, respectively. The size of the electrical RVE cube, which is suggested to be equal to or slightly larger than the average length of the CNT [26], was 0.6 μm in this case in order to balance the efficiency and the accuracy of the calculation. Regarding Case-2, only the nanocomposite with 0.5 wt% aligned MWCNTs was employed for the present validation. Tables 1 and 2 report the used parameters of the MWCNTs (grown by Baytubes C150P) [16,45] and PSF (UDEL P1700) [46]. The size of the electrical RVE cube was 1.0 μm for the same reason as Case-1. The results of the conductivity with different CNT position angles, ranging from 0� to 30� , and the comparison with experimental data for Case-1 are presented in Fig. 4(a) and (b). It should be noted that the minimal difference of the conductivity along longitudinal and transverse directions [14] was ignored and the average value was used in the present study. According to Fig. 4(a), the experimental conductivity shows good agreement with the results predicted from θmax ¼ 5� when the weight fraction is less than 2.0%. However, the conductivity can be accurately replicated by the model with θmax ¼ 30� when the CNT is 4.0 wt%. From the present simulated results, it can be concluded that the position angle is 5� when the CNT is 1.0 and 2.0 wt% while the angle is 30� if the fraction of the CNT is 4.0%. The TEM and SEM images shown in Ref. [14] also evidence the simulated results, indicating that it is hard Table 1 Properties of the MWCNT. Parameter
MWCNT for Case-1 [11,44]
MWCNT for Case-2 [16,45]
1.87
2.60
106
5 � 103
ΔE /eV
1
1
l /nm
600
1000
D /nm
10.7
12.0
M
460
460
dvdW /Å
3.4
3.4
1.4
1.4
E/GPa
2000
2000
ρMWCNT / g⋅cm gCNT / S⋅m
dcutoff /nm
Fig. 3. Validation of the electrical model (inserted: the validation on the percolation threshold of the present prediction). 4
1
3
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with the experimental measurement when the position angle is equal to 2� in the simulation for 0.5 wt% nanocomposite. All MWCNTs are aligned during the curing of matrix with the electrical field in Ref. [16]. Based on the results predicted in both Case-1 and Case-2, the position angle increases as the weight fraction of the CNTs increases. This behaviour can be explained by the fact that there is always a limitation for the alignment of CNTs when a high weight fraction is considered, independently of the technology used. With regards to the effect of the position angle on the conductivity, as shown in Fig. 4(b) and (c), the conductivity increases with the growth of the position angle for a fixed weight fraction following the same trend as reported in Ref. [7]. A larger position angle of CNT results in more possible conductive regions. As shown in Fig. 5, it is more likely for the CNTs to contact inside the matrix with a large position angle than per fect alignment, which leads to high conductivity in present work.
Table 2 Properties of the polymer. Parameter
ρepoxy / g⋅ cm gepoxy / S⋅ m
3 1
PET (Case-1) [10,14]
PSF (Case-2) [46]
1.41
1.24
~10
16
~10
E/GPa Yield stress (σy )/MPa
1.9 36
2.48 –
Strength (σs )/MPa
61.3
70.3
0.3
0.3
μ
15
to align and the increasing number (weight fraction) of CNTs because of the limited time spent on the aligning process. However, the aligning procedure introduced in Ref. [14] is suitable for this nanocomposite, especially when the CNT is less than 2.0 wt%. Additionally, for Case-2, the effect of the position angle, 0–5� , on the electric conductivity of the nanocomposite and comparison with the experimental data is presented in Fig. 4(c). It can be concluded that the results show a good agreement
Fig. 4. (a): Comparison on the conductivity between experimental and simulated results for Case-1; (b) effect of the position angle θmax on the conductivity obtained from simulations for Case-1; (c) effect of the position angle θmax on the conductivity from simulations and comparison with the experimental data for Case-2. 5
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Fig. 5. Schematic diagram for the conductive mechanism with small and large position angles.
3. Mechanical simulation
Table 3 Parameters of the FE models used in the present study.
3.1. FE model
Case ID
Based on the results of the electrical model (the position angle is 5� when the CNT is 1.0 and 2.0 wt% while the angle is 30� if the fraction of the CNT is 4.0% for Case-1 and position angle is equal to 2� for 0.5 wt% nanocomposite in Case-2), a large-scale FE model was built in the commercial software LS-DYNA®, which has the same size as the elec trical RVE (0:6 � 0:6 � 0:6 μm for Case-1 and 1:0 � 1:0 � 1:0 μm for Case-2). As the scheme shown in Fig. 6, a voxel mesh was generated with the software TexGEN® [47] to define an accurate contact behaviour and to reduce the distortion of the elements according to the previous work [48]. All the parameters of the FE models used in the present study are listed in Table 3. The mesh in the x-y plane is fine in the demand of reproducing the cross-section of the CNTs, where the smallest geometric dimension of the model, the diameter of the CNT, is presented. As for the boundary condition, in order to replicate a uniform stress state, the CONSTRAINED_NODE_SET in LS-DYNA® was employed to keep the deformation equal for the nodes on the same surface [48]. In Case-1, the loading direction was along the x-, y- and z-axial, respec tively, while the tension along the CNT-aligned direction (marked as
Case1 Case2
Weight fraction/wt.%
Position angle/degree
Number of CNTs
1.0 2.0 4.0 0.5
5 5 30 2
27 55 113 21
Number of elements (x � y � z) 100 � 100 � 50 120 � 120 � 50
z-axial) was loaded in Case-2. Regarding the modelling of interface be tween the CNT and the matrix, the tiebreak contact (with option 2) [49] was employed in LS-DYNA®. The interface along the normal direction was regarded as perfect bonding while the shear strength was 36 MPa, according to Ref. [18]. In order to investigate the effect of the bonding conditions of the interface on the mechanical properties of nano composite, perfect bonding (in any direction) was also modelled as a permanent tie (option 1 of the tiebreak contact [49]) in LS-DYNA®. 3.2. Material model The MAT_024 in LS-DYNA® was employed to describe the
Fig. 6. The geometry and the FE model used in the present study. 6
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improvement of the strength along transverse direction is potentially obtained due to the small cross-section of the CNTs. Similar results were also reported from other numerical frameworks [24]. However, the strength along the transverse direction obtained from tests is greater than the simulated strength. This increase in strength is potentially due to the transformation of the mechanical properties of the matrix by the addition of CNTs, which results in a rougher fracture surface formed after the failure of the matrix compared to the surface without CNTs [51, 52]. Furthermore, the combination between the CNT and crystalline polymer (PET in the present work) can also affect the strength. These combined effects lead to a different failure mechanism, which is difficult to describe in the numerical framework. Therefore, considering the transformation of the damage mechanism and the experimental error, the simulated results presented here are acceptable. Thus, the proposed approach, which utilizes the electric conductivity to determine the micro-structure of the nanocomposite, is reliable and can be employed to further investigate the effect of the loading direction, the interfacial strength and the weight fraction in the next section. The results from the simulation of the tensile tests and the comparison with the experimental data for Case-2 are presented in Fig. 8(d). The trend is similar to Case-1, i.e. with predicted results inside the two extremes. However, the experimental data lie closer to the simulation with NPI, indicating the poor interface between the MWCNTs and the amorphous polymer. Ac cording to the work of Klonos et al. [53], when the polymer is crystal line, it has a tiny effect on the electrical conductivity of nanocomposites. However, the mechanical property can be affected by the crystalline structure of the polymer. Related investigations [32,54] have shown the combination between the CNTs and crystalline polymer, leading to an outstanding interface property, i.e. high strength and even permanent stick combination, between the CNTs and polymer. That is also why the present results with a perfect interface can replicate the experiments in Case-1, while simulation with NPI can provide better results in Case-2.
Fig. 7. Validation of the material model for the neat PET.
mechanical response of the polymer, which contains both the elastic and plastic response. The basic mechanical parameters are listed in Table 2 for both cases. The plasticity of PET was reproduced by directly exploiting the experimental data provided in Ref. [14], while the trend of the neat PSF under tension was obtained from Ref. [50]. In order to validate the material model of the polymer, considering Case-1 as an example, the simulation of the PET under tensile loading was conducted on a unit cell, as shown in Fig. 7. The good agreement between the experimental and the simulated results clearly indicates that the mate rial model employed in the present study is reliable to describe the mechanical behaviour of the neat PET. Moreover, it should be noted that the strain at rupture was reported to be more than 300% in Ref. [14]. As a result, the failure strain of the neat PET was set to 300% in the present study, and the stress of the neat PET was kept constant after the drastic reduction from the peak stress. The dashed line in Fig. 7 presents the mechanical behaviour of the PET after the peak stress. Similar setting was applied on PSF according to Ref. [50]. Also, MAT_001 in LS-DYNA® was utilized to describe the perfectly elastic behaviour of CNTs and all the mechanical parameters used in this model are listed in Table 1 for both cases.
4. Analysis and discussion For further analysis, only Case-1 has been employed since the kind of samples are richer than those in Case-2. According to Fig. 8(a), (b) and (c), the variation of the stress-strain curves in the x and y directions is insignificant. Therefore, the simulated results from x and y directions can be combined, and only the x-direction loading is employed in the following discussion, identified as the transverse direction (TD). Mean while, the z direction is named as longitudinal direction (LD).
3.3. Model validation by comparison with experimental data
4.1. Differences in the loading condition
The tested results of the polymer/carbon nanotube samples described in Refs. [14,16] were conducted along different loading di rections: one direction was aligned to the CNT (z-direction in the present model, Fig. 6) and the transverse direction, respectively, in Case-1. In the numerical framework, the transverse direction was divided into the x- and y-direction to obtain more precise results. With regards to Case-2, only loading along the aligned direction was replicated as mentioned in Ref. [16]. The comparison of the numerical and experimental stress-strain curves along different directions of the samples for both cases is pre sented in Fig. 8. Regarding nanocomposites with different weight frac tions in Case-1, for low weight fractions (1.0 and 2.0 wt%), the experimental stress-strain curves along the z-direction present an in termediate behaviour and are thus positioned between the curves ob tained by the simulation with a perfect interface (PI), no failure on the interface, and with a non-perfect interface (NPI), modelled by tiebreak numerically, while the experimental stress-strain curves along the other directions are higher than the simulated ones (see Fig. 8(a) and (b)). When the weight fraction is equal to 4.0 wt%, a similar trend is visible in z-direction, but the stresses along the x-/y-direction obtained from simulations and experiments differ from each other. A limited
The difference between the two directions (DIFF) under varied weight fractions is presented in Fig. 9. Herein, DIFF is obtained by ðSLD STD Þ=SLD , where SLD and STD are the strengths along the LD and TD, respectively. It can be observed that the strength of LD is always slightly higher than that of TD, while a much larger DIFF was obtained with the PI, which reinforces the nanocomposite. The effect of the interface is discussed in more detail in the following section. The simulated results in 1.0 wt% with NPI are considered to analyse the difference between the loading conditions. The stress of the CNTs is always higher than 120 MPa when loading along the LD (see the red part in Fig. 10, bottom series). On the contrary, the CNTs and the matrix share a similar stress in the TD loading (see Fig. 10, top series). Ac cording to Fig. 10, the addition of CNTs changes the stress distribution: the stress of the matrix is lower near the CNTs, which is clearly visible under loading along the LD but secondary in the other case. It indicates that the CNTs carry the load leading to the significant increase of the strength when loading in the LD, showing the potentially outstanding mechanical properties of the nanocomposites.
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Fig. 8. Comparison of the simulated and experimentally obtained stress-strain curves along the x/y and z direction with 1.0 (a), 2.0 (b) and 4.0 (c) wt.% CNTs (the results from the models considering both the perfect and non-perfect interface are presented according to Section 3.1) for Case-1 and Case-2 (d).
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stress from the matrix to the CNTs causing low stress distributed on the matrix compared with the NPI, evidenced by the blue part in the left-top corner of the RVE in Fig. 11 (c), (d) with the PI where the CNTs are gathered. The outstanding ability on the stress transforming of the interface between the CNT and the crystalline structure is also one of the features of the nanocomposite with a crystalline polymer as a matrix. 4.3. Strength and modulus under different weight fraction Fig. 12 shows the change in modulus and strength with weight fraction of the CNTs from the simulated and the experimental results. Both strength and modulus along the LD increase with the increase of the weight fraction. As mentioned before, the effect of the interface on the modulus is limited. The focus therefore is put on the strength. Similar trends as experiments can be obtained with the NPI model, indicating the interface can be failed in present tests, but the interface should be stronger than the using NPI due to the gap between the experimental data and the predicted results, as shown in Fig. 12(b). Regarding the other direction, the simulated strengths and modulus along the TD are around 60 MPa, the strength of the neat PET shown in Table 2, with the change of the weight fraction. Because the effect of the CNTs on the strength in the TD is limited, as previously discussed in Section 4.1, the effect of the weight fraction on the mechanical properties in the TD is secondary. In addition, according to the spread of the experimental data re ported (see the variation bars in Fig. 12) [14], a larger variation is present in the results of 4.0 wt% (11.4% on the modulus and 4.7% on the strength) compared with those of 1.0 and 2.0 wt% (<3.2% on the modulus and <1.2% on the strength). The position angle, θmax , proposed in present study is for the random CNTs assigned within the nano composite, a larger position angle leads to more CNTs with different orientation and can thus cause a wide spread in the results, presented as the variation bar of the experimental data in Fig. 12. In the reported experimental data, spread found in the tests of 1.0 and 2.0 wt% nano composite, are reduced with respect to the spread of the results of 4.0 wt %, which evidences that the position angle of the 4.0 wt% should be larger than the other weight fractions, while 1.0 and 2.0 wt% nano composite might share a similar position angle due to the same spread observed in experiments. Such experimental data can validate the pre diction on the position angle obtained from the present electrical model: θmax ¼ 30� for 4.0 wt% and θmax ¼ 5� for the 1.0 and 2.0 wt% nanocomposite.
Fig. 9. The strength difference between two directions from experimental and simulated data with different weight fractions.
4.2. Effect of the interfacial strength In order to study the effect of the interfacial strength, perfect and non-perfect interface, were modelled by permanent tie and tiebreak namely in numerical models, the simulated results from the nano composite with 1.0 wt% CNTs were investigated. According to the stress-strain curves shown in Fig. 11 (a) and (b), the elastic modulus of the nanocomposite is practically unaffected by the interfacial strength due to the limited dimension of the CNTs (see the initiation of the curves in Fig. 11). However, the strength with the PI is always higher than that with the NPI. The PI has a considerable influence on the strength along the LD, while a small increase of strength is visible in case of the TD. When loading along the TD (Fig. 11 (c)), the PI and NPI show a similar stress field. As mentioned in Section 4.1, the addition of the CNTs has limited effects in the TD. Therefore, the interfacial strength is of minor importance in the reinforcement of the strength in the TD, while, on the contrary, it is of major importance in the LD. When the defor mation is larger than 2.02%, the stress on the CNTs is reduced with the NPI while they can still carry load with the PI (see Fig. 11 (d), after ε ¼ 2.02%). The bonding between the CNT and the matrix can transfer the
Fig. 10. Stress field when loading along the TD, x axis, (upper row) and LD, z axis, (lower row) (non-perfect interface with 1.0 wt% CNTs).
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Fig. 11. The effect of the interfacial strength on the stress-strain curve: TD (a), LD (b); and stress field: TD, load along x axis (c), LD, load along z axis (d) with 1.0 wt % CNTs.
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Fig. 12. The relationship between the weight fraction and the modulus (a)/strength (b) of the nanocomposite along the longitudinal direction (LD).
5. Conclusion and future research
on the electrical properties of nanocomposites reversely, as the present study, to extend the proposed approach from the aligned CNT system (the present cases) to randomly-distributed CNT system (real cases) and even other nanofillers. Overall, the proposed approach in the present work shows the powerful potential to numerically bridge the electrical and mechanical properties of nanocomposite.
A novel approach to determine the microstructure of a polymer/ carbon nanotube nanocomposite was developed to determine the me chanical property based on an electrical model. Firstly, an electrical model validated by the experimental results was used to characterize the position angle of the aligned carbon nanotube inside nanocomposite. Subsequently, a microstructural FE model of the RVE was built based on the prediction of the electrical model. To validate the present approach, two cases considering the crystalline structure of the matrix were involved in the present work. The stress-strain curves obtained from the FE model of the RVE and the experimental data of the aligned-MWCNT nanocomposite with semi-crystalline [14] and amorphous polymer [16] showed good agreement which demonstrates the reliability of the pro posed approach. Furthermore, the effect of the loading direction, the interfacial strength and the weight fraction of the CNTs was discussed in the numerical framework. The present work mainly achieves: (i) The mechanical properties of the nanocomposite (aligned CNTs) can be reliably replicated with the RVE FE model determined by the electrical property. The present method can be further validated by the matched stress-strain curves taking in account of the experimental spread. (ii) The difference on the mechanical property of nanocomposites with amorphous and crystalline polymer can be described through a changing interface property (nonperfect/perfect interface). (iii) The nanocomposite with the aligned CNTs can be regarded as a transversely isotropic material and CNTs provide a significant reinforcement of the mechanical properties along the longitudinal direction because it can carry load from the matrix, while a limited effect is presented in the transverse direction. (iv) A perfect interface improves the strength of the nanocomposite, especially along the longitudinal direction, while the effect on the modulus is almost negligible. (v) An increase of the weight fraction leads to the improvement of both the modulus and the strength. However, some factors may hinder this trend of the nanocomposite, such as the wavi ness, agglomeration and distribution of the CNTs inside the polymer. The proposed method may provide a novel approach to evaluate the mechanical property of nanocomposite and a novel method to investi gate nanocomposite’s mechanical behaviour. However, to reduce the complexity of the approach used to describe the microstructure, the proposed method was applied only to aligned-CNT nanocomposite in the present work. Actually, the waviness, agglomeration and distribution of CNTs can hardly be eliminated in most manufacturing of nano composite. Therefore, future research can be focused on investigations about the effect of the waviness, agglomeration and distribution of CNT
Declaration of competing interest This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue. The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript. CRediT authorship contribution statement Dayou Ma: Conceptualization, Methodology, Validation, Writing original draft. Marco Giglio: Project administration, Funding acquisi tion. Andrea Manes: Conceptualization, Writing - review & editing, Supervision, Project administration. Acknowledgements The author Dayou MA would like to thank the China Scholarship Council for their financial support (CSC, No. 201706290032). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.compscitech.2020.107993. References [1] Danikas MG. Nanocomposites – A Review of Electrical Treeing and Breakdown n. D.:19–25. [2] T. Tanaka, Dielectric Nanocomposites with Insulating Properties, 2005, pp. 914–928. [3] C. Li, E.T. Thostenson, T.W. Chou, Dominant role of tunneling resistance in the electrical conductivity of carbon nanotube-based composites, Appl. Phys. Lett. 91 (2007), https://doi.org/10.1063/1.2819690. [4] W.S. Bao, S.A. Meguid, Z.H. Zhu, Y. Pan, G.J. Weng, A novel approach to predict the electrical conductivity of multifunctional nanocomposites, Mech. Mater. 46 (2012) 129–138, https://doi.org/10.1016/J.MECHMAT.2011.12.006. [5] Z. Zabihi, H. Araghi, Monte Carlo simulations of effective electrical conductivity of graphene/poly ( methyl methacrylate ) nanocomposite : Landauer-Buttiker
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Contents lists available at ScienceDirect
Composites Science and Technology journal homepage: http://www.elsevier.com/locate/compscitech
An investigation into mechanical properties of the nanocomposite with aligned CNT by means of electrical conductivity Dayou Ma , Marco Giglio , Andrea Manes * Politecnico di Milano, Department of Mechanical Engineering, via la Masa, 1, 20156, Milan, Italy
A R T I C L E I N F O
A B S T R A C T
Keywords: Nano composites Electrical properties Interface Mechanical properties Finite element analysis (FEA)
In the present study, a novel modelling approach based on electrical properties was proposed to replicate the mechanical behaviour of aligned carbon nanotube/polymer nanocomposites. Firstly, an electrical analytical model with Monte-Carlo method involved was established and validated by accurately predicted electric con ductivity. The microstructure of the nanocomposite was then determined according to the electric property. Subsequently, a large-scale representative volume element model based on the predicted distribution of the carbon nanotubes was built to replicate the mechanical response of the nanocomposite under tension, which can be validated by existing experiments. To consider the crystalline structure of the matrix, two cases on the nanocomposites with crystalline and amorphous polymer were investigated, locating their difference on the bonding condition of the interface between CNT and matrix. Results evidenced that the electrical properties of nanocomposites can be used to identify the internal microstructure of nanocomposite. Moreover, the effects of the loading direction, the interfacial strength and the weight fraction were studied by numerical models. The reinforcement effect of the carbon nanotubes was significant when loaded along the aligned direction, but the effect was limited in the other directions. The modulus and the strength of nanocomposite were improved by the increase of the weight fraction of CNTs, while the increase of interfacial strength improves the strength of nanocomposite along CNT-aligned direction significantly, but had negligible effect on its modulus.
1. Introduction Nanocomposites have attracted great attentions in recent years due to their unique properties and potential applications. Among them, a polymer reinforced by carbon nanotube (CNT) is regarded as one of the most promising novel composites, since the reinforcement with CNTs not only improves the mechanical properties but also makes the com posites conductive, which provides a novel approach to investigate the mechanical behaviours of nanocomposites based on electrical properties. Considering the numerical and analytical analysis on the electrical properties of nanocomposites, the theory of electrical treeing and breakdown was proposed to describe the conductive mechanism of nanocomposites: a current path can be created among the CNTs without direct connections, which attributes to the tunnelling effect among CNTs [1]. Based on this, Tanaka [2] introduced a multicore model with the CNTs surrounded by multi-interface to describe the interaction zone between the CNTs and the matrix and it provided the proper electrical property of the nanocomposite. Li et al. [3] proposed a formula to
express the resistance of the tunnelling effect, which accelerated the development of the electrical analytical model. Meanwhile, Landauer-Buttiker (L-B) model, proposed to predict the resistance of the nanomaterials, and Monte-Carlo method, used to model the stochastic events, were proved to be helpful in electrical modelling [4,5]. Based on the aforementioned methods, the reliable electrical analytical model can be built for the investigation on the effects of the CNTs’ dimension, waviness and distribution on the electrical property of nanocomposites [6,7]. Briefly, all of these investigations indicate that a numerical methodology is able to accurately predict the electrical properties of nanocomposites. Regarding the mechanical properties of CNT nanocomposites, ex periments were mainly focused on tensile tests [8–16]. According to these experimental data, varying strength and modulus were reported in different studies but all confirmed the improvement of the modulus by the addition of CNTs. Besides, both strength and modulus can be rein forced when aligned CNTs are spread inside the polymer. As for the simulations, there are mainly two methods: a molecular dynamics (MD) simulation and a finite element (FE) method. The MD simulation is able to consider the chemical bonding and the related microstructure of the
* Corresponding author. E-mail addresses: [email protected] (D. Ma), [email protected] (M. Giglio), [email protected] (A. Manes). https://doi.org/10.1016/j.compscitech.2020.107993 Received 26 June 2019; Received in revised form 31 December 2019; Accepted 3 January 2020 Available online 8 January 2020 0266-3538/© 2020 Elsevier Ltd. All rights reserved.
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Nomenclature
Electron mass kg Polar angle � Maximum polar angle (position angle) � Azimuthal angle � The uniform random number in the interval [0,1] Intrinsic resistance of the CNT kΩ Contact resistance between the two closest CNTs kΩ Strength along the longitudinal direction MPa Strength along the transverse direction MPa Volume fraction of the nanocomposite Volume fraction of the percolation threshold Volume of object in percolation theory m3 Excluded volume m3 Transmission probability to breakdown the matrix between two CNTs ΔE The height of the barrier between polymer and CNT eV h Quantized resistance (� 12.9054) kΩ 2 2e ðxi ; yi ; zi Þ Coordinates of the start point of the CNT in model μm
me θ θmax φ RAND RCNT Rcontact SLD STD V Vc Vo Vex T
Symbol Description Units μ Poisson’s ratio D The diameter of the CNTs nm d Distance between two CNTs in the model nm dcutoff The cut-off distance in the tunnelling effect nm dtunnel The tunnelling characteristic length nm dvdW Van deer Waals separation distance Å e Electron charge E Elastic module GPa g Electric conductivity of the nanocomposite S/m gCNT Electric conductivity of the CNT S/m gpolymer Electric conductivity of the polymer S/m h The Planck’s constant (2πℏ ) J⋅s l The average length of the CNTs nm lij The length of two contact nodes i and j along one CNT μm Lx;y;z The length of the electrical RVE along the x-, y- & zdirection μm M Number of tunnelling channels
CNT-matrix debonding damage mechanism, which depends on the shear strength of the interface. Additionally, the failure mechanism of in terfaces also depends on the crystalline structure of the matrix [32]. Among these experimental studies, however, precise equipment is required and data are affected by a not negligible spread. Therefore, improvements about modelling methods are expected for a compre hensive understanding of mechanical response of nanocomposites. In present the study, in order to replicate the mechanical properties of the nanocomposite, a novel approach is proposed to model its microstructure based on the electrical conductivity. Because the microstructure can determine both the electrical and the mechanical behaviour of the polymer/Carbon nanotube nanocomposite, such an approach could be useful to predict one behaviour when knowing the other, especially when the nanocomposite is regular, such as aligned CNT employed in present study. Firstly, an electrical model, based on the previously introduced model in combination with the Monte-Carlo simulation, was built to replicate the electric conductivity of the nano composite by randomly distributed CNTs in the matrix. After the vali dation of this model, the microstructure of the nanocomposite with aligned CNTs can be obtained. Subsequently, a large-scale FE model was created to replicate the mechanical properties of this nanocomposite based on the results from the electrical model. The process is described in Fig. 1. Herein, the method was validated through the experimental data about tensile tests on the aligned-CNT/PET nanocomposite form [14] (named Case-1) to present the reliability of the proposed approach. Considering the crystalline structure of the polymer, one additional case, aligned-CNT/PSF nanocomposite from Ref. [16] (named Case-2), is also involved in the validation since PET is always regarded as a semi-crystalline polymer while PSF is a typical amorphous one. More over, the effect of the interfacial strength, the loading direction and the weight fraction is also discussed in a numerical framework.
nanocomposites. Therefore, the investigation of the interface between the CNTs and the matrix is always carried out in an MD simulation [17, 18]. Additionally, the effect of the CNTs dimension [19], posture [20] and distribution [21] on the mechanical property of nanocomposites has been investigated in this way, and their significant effects were proved. However, a long calculation time [20] and considerable parameters related to chemical bonds [22] cannot be avoided, hindering its wider application. Considering the FE method, the generation of a proper mesh is complex due to the dimensional gap between the CNTs and matrix. However, the FE method is still widely used on nanocomposites in the community due to its reliable modelling of complicated structures under various loading conditions and the user-friendly simulation environ ments. For instance, Chen and Liu [23] created a representative volume element (RVE) on the microscale to investigate the stiffness of nano composites. Arora et al. [24] studied the mechanical behaviour of nanocomposites along different directions and suggested to consider the material reinforced with aligned CNTs as a transversely isotropic ma terial. Also, Shin et al. [25] combined both the MD simulation and the FE method, which provided a new insight for the numerical study of nanocomposites. All of these existing studies, however, still have some limitations with regards to the modelling of the nanocomposites’ mechanical be haviours: the accurate microstructure of the nanocomposites and the failure mechanism of the interfaces between the CNT and matrix. Only one/a few CNT(s) can be contained in the numerical model because of computational costs [17,18,20–22], or random generation was employed to build the CNT system inside nanocomposites [4,6,7,26]. Although the Monte-Carlo simulation can provide good results through the random generations, extensive calculations are inevitable. Efforts have been made on modelling the microstructure of nanocomposites according to the images of sample cross-sections [27,28]. However, the accuracy might be affected due to the possibility that the real structure cannot be represented by a single cross-section and the cost of this technology presents a further drawback. Therefore, large-scale model ling, especially containing a precise microstructure, is still a challenge in the simulation of nanocomposites [29]. As for investigation about the failure mechanism of interfaces between the CNT and the matrix, Yu et al. [30] combined a loading device and scanning electron microscopy (SEM), while Cooper et al. [31] studied the tension response of the nanocomposite with post analysis by transmission electron microscopy (TEM). They reported that the “pull-out” failure of CNTs is the dominant
2. Electrical simulations 2.1. Modelling strategy In the present study, all CNTs are assumed to be straight to model aligned CNTs, therefore a line segment can be used to model the CNT. Considering the random distribution of the CNTs inside the matrix, the start point of the CNT ðxi ; yi ; zi Þ, as shown in Fig. 2, is generated ac cording to Eq. (1). 2
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Fig. 1. flowchart for present method.
Fig. 2. A schematic diagram of the CNT inside the nanocomposite.
perfect circle with the diameter D. lij is the length of two contact nodes i and j along one CNT. The contact resistance, Rcontact , caused by con tacting between CNTs and the tunnelling effect. According to L-B model [34], Rcontact is defined by Eq. (5), where 2eh2 is equal to the quantized resistance (� 12:9054kΩ). Moreover, the transmission probability to break down the matrix between two CNTs, T, is determined by Eq. (6) based on Wentzel-Kramers-Brillouin (WKB) approximation [35]. With the Pauli exclusion principle [36], the closest distance between two CNTs, d, should be no less than the van deer Waals separation distance, dvdW . So, if d is smaller than dvdW , it should be replaced by dvdW , which is used to model the resistance due to contact between the CNTs. When d is greater than dvdW , the real distance is equal to (d D) considering the diameter of the CNT, i.e. the tunnelling effect is built in this way. But the tunnelling effect disappears if the distance is beyond the cut-off distance, dcutoff , then there is no effective current path any more. Moreover, dtunnel pffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the tunnelling characteristic length defined by dtunnel ¼ ℏ= 8me ΔE.
(1)
Θi ¼ LΘ � RANDðΘ ¼ x; y; z:i ¼ 1; 2; …Þ
A CNT built as a line segment and the coordinate system generated based on the CNT are shown in Fig. 2, θ is the polar angle, between the CNT and the z-axial, defined by the maximum polar angle θmax (Eq. (2)), while the azimuthal angle φ, the angle between x-axial and the projec tion of the CNT on the x-y plane, is generated inside [0, 2π] randomly (Eq. (3)). The θmax represents the position angle in present study, which determines the maximum polar angle of all CNTs. If θmax ¼ 0, the nanocomposite is assembled with all CNTs aligned along one direction. If θmax ¼ 90� , the nanocomposite can be regarded as an isotropic ma terial, which indicates the totally random distribution of aligned CNTs in the matrix. cosθ ¼ ð1
(2)
cosθmax Þ � RAND þ cosθmax
(3)
φ ¼ 2π � RAND
As for the length of each CNT, an average length l of all CNTs was used in the present study. The average length can always be obtained by atomic force microscopy and the length of the CNTs should follow the Weibull distribution [33]. Subsequently, the coordinates of the end point of ith CNT can be described as (xi þ lsinθi cosφi ; yi þ lsinθi sinφi ; zi þ 0
RCNT ¼
4lij
π gCNT D2
Rcontact ¼
lcosθi ). Moreover, if the end point is out of the boundary of electrical RVE, the CNT would be cut directly by the boundary. In this way, a random CNT network can be generated in the electrical RVE. Two types of electric resistance existing in this CNT system contribute to the total resistance of the nanocomposite. One is the intrinsic resistance of the CNT, RCNT , and the other is the contact resis tance between the two closest CNTs, Rcontact . RCNT can be obtained through Eq. (4), where the cross section of the CNT is regarded as a
h 1 2e2 MT
� � dvdW exp ; 0 � d � D þ dvdW dtunnel T¼ � � > d D > > : exp ; D þ dvdW � d � D þ dcutoff dtunnel 8 > > > <
(4) (5)
(6)
In the present study, the distance between every two CNTs was calculated through the methodology mentioned in Ref. [37] providing a matrice of distance between every two CNTs in the network. Then, the 3
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matrices of all potential resistances can be calculated by Eqs. (4)–(6). Subsequently, a searching algorithm was employed to obtain the effec tive current method in the CNT system and all of the electric resistance from the effective current paths are regarded as being connected in parallel. Additionally, the Monte-Carlo simulation was involved in this electrical model for each case. The above-mentioned process was per formed in MATLAB®.
provided a possible method to link the percolation threshold to the aspect ratio of the CNT based on the statistical percolation theory. Meanwhile, the DPD (dissipative particle dynamics) method coupled with Monte-Carlo simulations is also able to replicate the percolation threshold [43] but requires time-consuming calculation. Herein, we employed the statistical percolation theory to obtain the percolation threshold and compared it with the existing data to validate the present electrical model. Based on the statistical percolation theory with t ¼ 1:414, the percolation threshold is approximately 0.62 vol%. With these parameters, the linear relationship of the logðgÞ and logðV Vc Þ is pre sented in Fig. 3, as a validation. The value of the percolation threshold predicted in the present work stands in good agreement with the threshold predicted in Refs. [7,28], while the value theoretically pre dicted by the aspect ratio of CNTs [42] is 0.631 vol%. Therefore, the present electrical model was validated and can be used in the next stage. In the present study, the electrical model introduced in this section was employed to obtain the position angle θmax exploiting a reverse engi neering approach.
2.2. Validation of the electrical model To validate the electrical model, the electric conductivity of a multiwall CNT (MWCNT) distributed in the polymer was calculated. All the data used in present model were taken from Refs. [7,38]: gCNT ¼ 5� 103 S/m, gpolymer ¼ 10 12 S/m, ΔE ¼ 1 eV, l ¼ 5 μm, D ¼ 50 nm, M ¼ 460, Lx ¼ Ly ¼ Lz ¼ 5:5 μm, θmax ¼ 2π , dvdW ¼ 3:4 Å and dcutoff ¼ 1:4 nm. The increasing trend of conductivity with volume fraction is shown in Fig. 3. The results obtained from the present model correspond to the existing experimental data [7,28]. The electrical conductivity steadily increases while its growing rate decreases with the increasing weight fraction of the CNTs, indicating the gradual disappearance of the tunnelling effect. As the tunnelling effect plays the dominant role in generating the effective current when the volume fraction is low, the electrical conductivity is sensitive to the number of the CNTs in this region. On the contrary, the increase of the electrical conductivity is no longer significant after reaching the percolation threshold, which is the saturation point of the tunnelling effect, as shown in Fig. 3. There are different theoretical methods to predict the percolation threshold of nanocomposite. The theory related to the concept of excluded volume (Vex ), which is determined as the volume around the centre of one object which doesn’t allow others to penetrate, is reported effective [39]. The percolation threshold can be expressed by the excluded volume as Vc � Vo =hVex i [40], where h i is the average calculation in a random-distributed system, Vo is the volume of the object and Vc is the percolation threshold represented in the volume fraction. In addition, based on the data of the electric conductivity, the statistical percolation theory can also provide accurate results, which can be expressed by describing the relationship between the existing data about the volume fraction (V) and the electric conductivity (g) using g ¼ A⋅ ðV Vc Þt . The parameters A and t are defined by the data when the logðgÞ and logðV Vc Þ are linear-related. However, t is proven to be related with the material experimentally [41], while A can also be calculated through the conductivity of the nanofillers [42]. Moreover, the work of [42] also
2.3. Simulation results In this section, the position angle θmax of the polyethylene tere phthalate/MWCNT (PET/MWCNT, Case-1) and polysulfone/MWCNT (PSF/MWCNT, Case-2) were determined based on the electric conduc tivity reported in Refs. [14,16], respectively. For Case-1, the CNTs were dispersed into PET and then aligned along one direction (marked as z-direction in the present work) during the manufacturing process. Three weight fractions of CNTs, 1.0%, 2.0%, and 4.0%, were investi gated. Relative properties of the MWCNT (Nanocyl® 7000) [11,44] and PET (Polyclear F019) [10,14] are listed in Table 1 and Table 2, respectively. The size of the electrical RVE cube, which is suggested to be equal to or slightly larger than the average length of the CNT [26], was 0.6 μm in this case in order to balance the efficiency and the accuracy of the calculation. Regarding Case-2, only the nanocomposite with 0.5 wt% aligned MWCNTs was employed for the present validation. Tables 1 and 2 report the used parameters of the MWCNTs (grown by Baytubes C150P) [16,45] and PSF (UDEL P1700) [46]. The size of the electrical RVE cube was 1.0 μm for the same reason as Case-1. The results of the conductivity with different CNT position angles, ranging from 0� to 30� , and the comparison with experimental data for Case-1 are presented in Fig. 4(a) and (b). It should be noted that the minimal difference of the conductivity along longitudinal and transverse directions [14] was ignored and the average value was used in the present study. According to Fig. 4(a), the experimental conductivity shows good agreement with the results predicted from θmax ¼ 5� when the weight fraction is less than 2.0%. However, the conductivity can be accurately replicated by the model with θmax ¼ 30� when the CNT is 4.0 wt%. From the present simulated results, it can be concluded that the position angle is 5� when the CNT is 1.0 and 2.0 wt% while the angle is 30� if the fraction of the CNT is 4.0%. The TEM and SEM images shown in Ref. [14] also evidence the simulated results, indicating that it is hard Table 1 Properties of the MWCNT. Parameter
MWCNT for Case-1 [11,44]
MWCNT for Case-2 [16,45]
1.87
2.60
106
5 � 103
ΔE /eV
1
1
l /nm
600
1000
D /nm
10.7
12.0
M
460
460
dvdW /Å
3.4
3.4
1.4
1.4
E/GPa
2000
2000
ρMWCNT / g⋅cm gCNT / S⋅m
dcutoff /nm
Fig. 3. Validation of the electrical model (inserted: the validation on the percolation threshold of the present prediction). 4
1
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with the experimental measurement when the position angle is equal to 2� in the simulation for 0.5 wt% nanocomposite. All MWCNTs are aligned during the curing of matrix with the electrical field in Ref. [16]. Based on the results predicted in both Case-1 and Case-2, the position angle increases as the weight fraction of the CNTs increases. This behaviour can be explained by the fact that there is always a limitation for the alignment of CNTs when a high weight fraction is considered, independently of the technology used. With regards to the effect of the position angle on the conductivity, as shown in Fig. 4(b) and (c), the conductivity increases with the growth of the position angle for a fixed weight fraction following the same trend as reported in Ref. [7]. A larger position angle of CNT results in more possible conductive regions. As shown in Fig. 5, it is more likely for the CNTs to contact inside the matrix with a large position angle than per fect alignment, which leads to high conductivity in present work.
Table 2 Properties of the polymer. Parameter
ρepoxy / g⋅ cm gepoxy / S⋅ m
3 1
PET (Case-1) [10,14]
PSF (Case-2) [46]
1.41
1.24
~10
16
~10
E/GPa Yield stress (σy )/MPa
1.9 36
2.48 –
Strength (σs )/MPa
61.3
70.3
0.3
0.3
μ
15
to align and the increasing number (weight fraction) of CNTs because of the limited time spent on the aligning process. However, the aligning procedure introduced in Ref. [14] is suitable for this nanocomposite, especially when the CNT is less than 2.0 wt%. Additionally, for Case-2, the effect of the position angle, 0–5� , on the electric conductivity of the nanocomposite and comparison with the experimental data is presented in Fig. 4(c). It can be concluded that the results show a good agreement
Fig. 4. (a): Comparison on the conductivity between experimental and simulated results for Case-1; (b) effect of the position angle θmax on the conductivity obtained from simulations for Case-1; (c) effect of the position angle θmax on the conductivity from simulations and comparison with the experimental data for Case-2. 5
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Fig. 5. Schematic diagram for the conductive mechanism with small and large position angles.
3. Mechanical simulation
Table 3 Parameters of the FE models used in the present study.
3.1. FE model
Case ID
Based on the results of the electrical model (the position angle is 5� when the CNT is 1.0 and 2.0 wt% while the angle is 30� if the fraction of the CNT is 4.0% for Case-1 and position angle is equal to 2� for 0.5 wt% nanocomposite in Case-2), a large-scale FE model was built in the commercial software LS-DYNA®, which has the same size as the elec trical RVE (0:6 � 0:6 � 0:6 μm for Case-1 and 1:0 � 1:0 � 1:0 μm for Case-2). As the scheme shown in Fig. 6, a voxel mesh was generated with the software TexGEN® [47] to define an accurate contact behaviour and to reduce the distortion of the elements according to the previous work [48]. All the parameters of the FE models used in the present study are listed in Table 3. The mesh in the x-y plane is fine in the demand of reproducing the cross-section of the CNTs, where the smallest geometric dimension of the model, the diameter of the CNT, is presented. As for the boundary condition, in order to replicate a uniform stress state, the CONSTRAINED_NODE_SET in LS-DYNA® was employed to keep the deformation equal for the nodes on the same surface [48]. In Case-1, the loading direction was along the x-, y- and z-axial, respec tively, while the tension along the CNT-aligned direction (marked as
Case1 Case2
Weight fraction/wt.%
Position angle/degree
Number of CNTs
1.0 2.0 4.0 0.5
5 5 30 2
27 55 113 21
Number of elements (x � y � z) 100 � 100 � 50 120 � 120 � 50
z-axial) was loaded in Case-2. Regarding the modelling of interface be tween the CNT and the matrix, the tiebreak contact (with option 2) [49] was employed in LS-DYNA®. The interface along the normal direction was regarded as perfect bonding while the shear strength was 36 MPa, according to Ref. [18]. In order to investigate the effect of the bonding conditions of the interface on the mechanical properties of nano composite, perfect bonding (in any direction) was also modelled as a permanent tie (option 1 of the tiebreak contact [49]) in LS-DYNA®. 3.2. Material model The MAT_024 in LS-DYNA® was employed to describe the
Fig. 6. The geometry and the FE model used in the present study. 6
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improvement of the strength along transverse direction is potentially obtained due to the small cross-section of the CNTs. Similar results were also reported from other numerical frameworks [24]. However, the strength along the transverse direction obtained from tests is greater than the simulated strength. This increase in strength is potentially due to the transformation of the mechanical properties of the matrix by the addition of CNTs, which results in a rougher fracture surface formed after the failure of the matrix compared to the surface without CNTs [51, 52]. Furthermore, the combination between the CNT and crystalline polymer (PET in the present work) can also affect the strength. These combined effects lead to a different failure mechanism, which is difficult to describe in the numerical framework. Therefore, considering the transformation of the damage mechanism and the experimental error, the simulated results presented here are acceptable. Thus, the proposed approach, which utilizes the electric conductivity to determine the micro-structure of the nanocomposite, is reliable and can be employed to further investigate the effect of the loading direction, the interfacial strength and the weight fraction in the next section. The results from the simulation of the tensile tests and the comparison with the experimental data for Case-2 are presented in Fig. 8(d). The trend is similar to Case-1, i.e. with predicted results inside the two extremes. However, the experimental data lie closer to the simulation with NPI, indicating the poor interface between the MWCNTs and the amorphous polymer. Ac cording to the work of Klonos et al. [53], when the polymer is crystal line, it has a tiny effect on the electrical conductivity of nanocomposites. However, the mechanical property can be affected by the crystalline structure of the polymer. Related investigations [32,54] have shown the combination between the CNTs and crystalline polymer, leading to an outstanding interface property, i.e. high strength and even permanent stick combination, between the CNTs and polymer. That is also why the present results with a perfect interface can replicate the experiments in Case-1, while simulation with NPI can provide better results in Case-2.
Fig. 7. Validation of the material model for the neat PET.
mechanical response of the polymer, which contains both the elastic and plastic response. The basic mechanical parameters are listed in Table 2 for both cases. The plasticity of PET was reproduced by directly exploiting the experimental data provided in Ref. [14], while the trend of the neat PSF under tension was obtained from Ref. [50]. In order to validate the material model of the polymer, considering Case-1 as an example, the simulation of the PET under tensile loading was conducted on a unit cell, as shown in Fig. 7. The good agreement between the experimental and the simulated results clearly indicates that the mate rial model employed in the present study is reliable to describe the mechanical behaviour of the neat PET. Moreover, it should be noted that the strain at rupture was reported to be more than 300% in Ref. [14]. As a result, the failure strain of the neat PET was set to 300% in the present study, and the stress of the neat PET was kept constant after the drastic reduction from the peak stress. The dashed line in Fig. 7 presents the mechanical behaviour of the PET after the peak stress. Similar setting was applied on PSF according to Ref. [50]. Also, MAT_001 in LS-DYNA® was utilized to describe the perfectly elastic behaviour of CNTs and all the mechanical parameters used in this model are listed in Table 1 for both cases.
4. Analysis and discussion For further analysis, only Case-1 has been employed since the kind of samples are richer than those in Case-2. According to Fig. 8(a), (b) and (c), the variation of the stress-strain curves in the x and y directions is insignificant. Therefore, the simulated results from x and y directions can be combined, and only the x-direction loading is employed in the following discussion, identified as the transverse direction (TD). Mean while, the z direction is named as longitudinal direction (LD).
3.3. Model validation by comparison with experimental data
4.1. Differences in the loading condition
The tested results of the polymer/carbon nanotube samples described in Refs. [14,16] were conducted along different loading di rections: one direction was aligned to the CNT (z-direction in the present model, Fig. 6) and the transverse direction, respectively, in Case-1. In the numerical framework, the transverse direction was divided into the x- and y-direction to obtain more precise results. With regards to Case-2, only loading along the aligned direction was replicated as mentioned in Ref. [16]. The comparison of the numerical and experimental stress-strain curves along different directions of the samples for both cases is pre sented in Fig. 8. Regarding nanocomposites with different weight frac tions in Case-1, for low weight fractions (1.0 and 2.0 wt%), the experimental stress-strain curves along the z-direction present an in termediate behaviour and are thus positioned between the curves ob tained by the simulation with a perfect interface (PI), no failure on the interface, and with a non-perfect interface (NPI), modelled by tiebreak numerically, while the experimental stress-strain curves along the other directions are higher than the simulated ones (see Fig. 8(a) and (b)). When the weight fraction is equal to 4.0 wt%, a similar trend is visible in z-direction, but the stresses along the x-/y-direction obtained from simulations and experiments differ from each other. A limited
The difference between the two directions (DIFF) under varied weight fractions is presented in Fig. 9. Herein, DIFF is obtained by ðSLD STD Þ=SLD , where SLD and STD are the strengths along the LD and TD, respectively. It can be observed that the strength of LD is always slightly higher than that of TD, while a much larger DIFF was obtained with the PI, which reinforces the nanocomposite. The effect of the interface is discussed in more detail in the following section. The simulated results in 1.0 wt% with NPI are considered to analyse the difference between the loading conditions. The stress of the CNTs is always higher than 120 MPa when loading along the LD (see the red part in Fig. 10, bottom series). On the contrary, the CNTs and the matrix share a similar stress in the TD loading (see Fig. 10, top series). Ac cording to Fig. 10, the addition of CNTs changes the stress distribution: the stress of the matrix is lower near the CNTs, which is clearly visible under loading along the LD but secondary in the other case. It indicates that the CNTs carry the load leading to the significant increase of the strength when loading in the LD, showing the potentially outstanding mechanical properties of the nanocomposites.
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Fig. 8. Comparison of the simulated and experimentally obtained stress-strain curves along the x/y and z direction with 1.0 (a), 2.0 (b) and 4.0 (c) wt.% CNTs (the results from the models considering both the perfect and non-perfect interface are presented according to Section 3.1) for Case-1 and Case-2 (d).
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stress from the matrix to the CNTs causing low stress distributed on the matrix compared with the NPI, evidenced by the blue part in the left-top corner of the RVE in Fig. 11 (c), (d) with the PI where the CNTs are gathered. The outstanding ability on the stress transforming of the interface between the CNT and the crystalline structure is also one of the features of the nanocomposite with a crystalline polymer as a matrix. 4.3. Strength and modulus under different weight fraction Fig. 12 shows the change in modulus and strength with weight fraction of the CNTs from the simulated and the experimental results. Both strength and modulus along the LD increase with the increase of the weight fraction. As mentioned before, the effect of the interface on the modulus is limited. The focus therefore is put on the strength. Similar trends as experiments can be obtained with the NPI model, indicating the interface can be failed in present tests, but the interface should be stronger than the using NPI due to the gap between the experimental data and the predicted results, as shown in Fig. 12(b). Regarding the other direction, the simulated strengths and modulus along the TD are around 60 MPa, the strength of the neat PET shown in Table 2, with the change of the weight fraction. Because the effect of the CNTs on the strength in the TD is limited, as previously discussed in Section 4.1, the effect of the weight fraction on the mechanical properties in the TD is secondary. In addition, according to the spread of the experimental data re ported (see the variation bars in Fig. 12) [14], a larger variation is present in the results of 4.0 wt% (11.4% on the modulus and 4.7% on the strength) compared with those of 1.0 and 2.0 wt% (<3.2% on the modulus and <1.2% on the strength). The position angle, θmax , proposed in present study is for the random CNTs assigned within the nano composite, a larger position angle leads to more CNTs with different orientation and can thus cause a wide spread in the results, presented as the variation bar of the experimental data in Fig. 12. In the reported experimental data, spread found in the tests of 1.0 and 2.0 wt% nano composite, are reduced with respect to the spread of the results of 4.0 wt %, which evidences that the position angle of the 4.0 wt% should be larger than the other weight fractions, while 1.0 and 2.0 wt% nano composite might share a similar position angle due to the same spread observed in experiments. Such experimental data can validate the pre diction on the position angle obtained from the present electrical model: θmax ¼ 30� for 4.0 wt% and θmax ¼ 5� for the 1.0 and 2.0 wt% nanocomposite.
Fig. 9. The strength difference between two directions from experimental and simulated data with different weight fractions.
4.2. Effect of the interfacial strength In order to study the effect of the interfacial strength, perfect and non-perfect interface, were modelled by permanent tie and tiebreak namely in numerical models, the simulated results from the nano composite with 1.0 wt% CNTs were investigated. According to the stress-strain curves shown in Fig. 11 (a) and (b), the elastic modulus of the nanocomposite is practically unaffected by the interfacial strength due to the limited dimension of the CNTs (see the initiation of the curves in Fig. 11). However, the strength with the PI is always higher than that with the NPI. The PI has a considerable influence on the strength along the LD, while a small increase of strength is visible in case of the TD. When loading along the TD (Fig. 11 (c)), the PI and NPI show a similar stress field. As mentioned in Section 4.1, the addition of the CNTs has limited effects in the TD. Therefore, the interfacial strength is of minor importance in the reinforcement of the strength in the TD, while, on the contrary, it is of major importance in the LD. When the defor mation is larger than 2.02%, the stress on the CNTs is reduced with the NPI while they can still carry load with the PI (see Fig. 11 (d), after ε ¼ 2.02%). The bonding between the CNT and the matrix can transfer the
Fig. 10. Stress field when loading along the TD, x axis, (upper row) and LD, z axis, (lower row) (non-perfect interface with 1.0 wt% CNTs).
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Fig. 11. The effect of the interfacial strength on the stress-strain curve: TD (a), LD (b); and stress field: TD, load along x axis (c), LD, load along z axis (d) with 1.0 wt % CNTs.
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Fig. 12. The relationship between the weight fraction and the modulus (a)/strength (b) of the nanocomposite along the longitudinal direction (LD).
5. Conclusion and future research
on the electrical properties of nanocomposites reversely, as the present study, to extend the proposed approach from the aligned CNT system (the present cases) to randomly-distributed CNT system (real cases) and even other nanofillers. Overall, the proposed approach in the present work shows the powerful potential to numerically bridge the electrical and mechanical properties of nanocomposite.
A novel approach to determine the microstructure of a polymer/ carbon nanotube nanocomposite was developed to determine the me chanical property based on an electrical model. Firstly, an electrical model validated by the experimental results was used to characterize the position angle of the aligned carbon nanotube inside nanocomposite. Subsequently, a microstructural FE model of the RVE was built based on the prediction of the electrical model. To validate the present approach, two cases considering the crystalline structure of the matrix were involved in the present work. The stress-strain curves obtained from the FE model of the RVE and the experimental data of the aligned-MWCNT nanocomposite with semi-crystalline [14] and amorphous polymer [16] showed good agreement which demonstrates the reliability of the pro posed approach. Furthermore, the effect of the loading direction, the interfacial strength and the weight fraction of the CNTs was discussed in the numerical framework. The present work mainly achieves: (i) The mechanical properties of the nanocomposite (aligned CNTs) can be reliably replicated with the RVE FE model determined by the electrical property. The present method can be further validated by the matched stress-strain curves taking in account of the experimental spread. (ii) The difference on the mechanical property of nanocomposites with amorphous and crystalline polymer can be described through a changing interface property (nonperfect/perfect interface). (iii) The nanocomposite with the aligned CNTs can be regarded as a transversely isotropic material and CNTs provide a significant reinforcement of the mechanical properties along the longitudinal direction because it can carry load from the matrix, while a limited effect is presented in the transverse direction. (iv) A perfect interface improves the strength of the nanocomposite, especially along the longitudinal direction, while the effect on the modulus is almost negligible. (v) An increase of the weight fraction leads to the improvement of both the modulus and the strength. However, some factors may hinder this trend of the nanocomposite, such as the wavi ness, agglomeration and distribution of the CNTs inside the polymer. The proposed method may provide a novel approach to evaluate the mechanical property of nanocomposite and a novel method to investi gate nanocomposite’s mechanical behaviour. However, to reduce the complexity of the approach used to describe the microstructure, the proposed method was applied only to aligned-CNT nanocomposite in the present work. Actually, the waviness, agglomeration and distribution of CNTs can hardly be eliminated in most manufacturing of nano composite. Therefore, future research can be focused on investigations about the effect of the waviness, agglomeration and distribution of CNT
Declaration of competing interest This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue. The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript. CRediT authorship contribution statement Dayou Ma: Conceptualization, Methodology, Validation, Writing original draft. Marco Giglio: Project administration, Funding acquisi tion. Andrea Manes: Conceptualization, Writing - review & editing, Supervision, Project administration. Acknowledgements The author Dayou MA would like to thank the China Scholarship Council for their financial support (CSC, No. 201706290032). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.compscitech.2020.107993. References [1] Danikas MG. Nanocomposites – A Review of Electrical Treeing and Breakdown n. D.:19–25. [2] T. Tanaka, Dielectric Nanocomposites with Insulating Properties, 2005, pp. 914–928. [3] C. Li, E.T. Thostenson, T.W. Chou, Dominant role of tunneling resistance in the electrical conductivity of carbon nanotube-based composites, Appl. Phys. Lett. 91 (2007), https://doi.org/10.1063/1.2819690. [4] W.S. Bao, S.A. Meguid, Z.H. Zhu, Y. Pan, G.J. Weng, A novel approach to predict the electrical conductivity of multifunctional nanocomposites, Mech. Mater. 46 (2012) 129–138, https://doi.org/10.1016/J.MECHMAT.2011.12.006. [5] Z. Zabihi, H. Araghi, Monte Carlo simulations of effective electrical conductivity of graphene/poly ( methyl methacrylate ) nanocomposite : Landauer-Buttiker
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